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\title{《Mathematical Writing \\ Chapter 3: Mathematical Writing}
%(1.1-1.2) 
%\institute{上海立信会计金融学院}
\author{NJH}
%\date{}

\maketitle

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\begin{frame}{Contents }

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\begin{myenumerate}
\item What is a Theorem?
\item Proofs
\item The Role of Examples
\item Definitions
\item Notation
\item Words versus Symbols
\item Displaying Equations
\item Parallelism
\item Dos and Don'ts 
\item Miscellaneous
\end{myenumerate}

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\begin{frame}{Quotations 1 }

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\begin{myitemize}
\item Suppose you want to teach the ``cat'' concept to a very young child.

\item Do you explain that a cat is a relatively small,
primarily carnivorous mammal with retractile claws,
a distinctive sonic output, etc.?

\item I'll bet not.

\pause 

\item You probably show the kid a lot of different cats,
saying ``kitty'' each time, until it gets the idea.

\item {\bf\color{red}To put it more generally, generalizations are best made by abstraction from experience.}

\hfill -- Ralph P. Boas, Can We Make Mathematics Intelligible? (1981)

%\hfill -- RALPH P. BOAS, Can We Make Mathematics Intelligible? (1981)

\end{myitemize}

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\begin{frame}{Quotations 2 - 4 }

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\begin{myitemize}
\item {\bf\color{red}A good notation should be unambiguous, pregnant, easy to remember;
it should avoid harmful second meanings, and take advantage of useful second meanings;}
the order and connection of signs should suggest the order and connection of things.

\hfill -- George Polya, How to Solve It (1957)
%\hfill - GEORGE POLYA , How to Solve It (1957)

\pause 

\item We have not succeeded in finding or constructing a definition which starts out
``A Bravais lattice is $\cdots$''; the sources we have looked at say ``That was a Bravais lattice.''

\hfill -- Charles Kittel, Introduction to Solid State Physics (1971)
%\hfill - CHARLES KITTEL, Introduction to Solid State Physics (1971)

\pause 

\item {\bf\color{red}Notation is everything.}

\hfill -- Charles F. Van Loan, FFTs and the Sparse Factorization Idea (1992)
%\hfill - CHARLES F. VAN LOAN, FFTs and the Sparse Factorization Idea (1992)

\end{myitemize}

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%\begin{frame}{Introduction }
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%\begin{myitemize}
%\item The mathematical writer needs to be aware of a number of matters specific to mathematical writing, ranging from general issues, such as choice of notation, to particular details, such as how to punctuate mathematical expressions. 
%
%\item In this chapter I begin by discussing some of the general issues and then move on to specifics.
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%\end{myitemize}
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\begin{frame}{ What is a Theorem? - 1 }

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\begin{myitemize}
\item  What are the differences between theorems, lemmas, and propositions?
To some extent, the answer depends on the context in which a result appears.

\pause 

\item  {\bf\color{red}Generally, a theorem is a major result that is of independent interest.
The proof of a theorem is usually nontrivial.} A lemma is an auxiliary result
-- a stepping stone towards a theorem.
Its proof may be easy or difficult.

\pause 

\item A straightforward and independent result that is worth encapsulating but that does not merit the title of a theorem may also be called a lemma. 

\pause 

\item Indeed, there are some famous lemmas, such as the Riemann-Lebesgue Lemma in the theory of Fourier series and Farkas's Lemma in the theory of constrained optimization.

\end{myitemize}

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\begin{frame}{ What is a Theorem? - 2 }

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\begin{myitemize}
\item Whether a result should be stated formally as a lemma or simply mentioned in the text depends on the level at which you are writing.

\pause 


\item  {\bf\color{red}In a research paper in linear algebra it would be inappropriate to give a lemma stating that the eigenvalues of a \underline{symmetric positive definite matrix} are positive, as this standard result is so well known; but in a textbook for undergraduates it would be sensible to formalize this result.}

\pause 


\item  It is not advisable to label all your results theorems, because if you do so you miss the opportunity to emphasize the logical structure of your work and to direct attention to the most important results.
If you are in doubt about whether to call a result a lemma or a theorem, call it a lemma.
  
\end{myitemize}

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\begin{frame}{ What is a Theorem? - 3 }

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\begin{myitemize}
\item  {\bf\color{red}The term proposition is less widely used than lemma and theorem and its meaning is less clear.
It tends to be used as a way to denote a minor theorem.}

\pause 


\item  Lecturers and textbook authors might feel that the modest tone of its name makes a proposition appear less daunting to students than a theorem. However, a proposition is not, as one student thought, ``a theorem that might not be true''.

\pause 


\item  {\bf\color{red}A corollary is a direct or easy consequence of a lemma, theorem or proposition.}
It is important to distinguish between a corollary, which does not imply the parent result from which it came, and an extension or generalization of a result.

\end{myitemize}

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\begin{frame}{ What is a Theorem? - 4  }

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\begin{myitemize}
\item Be careful not to over-glorify a corollary by failing to label it as such, for this gives it false prominence and obscures the role of the parent result. The plural of lemma is lemmata, or, more commonly, lemmas.

\pause 

\item  {\bf\color{red}How many results are formally stated as lemmas, theorems, propositions or corollaries is a matter of personal style.} Some authors develop their ideas in a sequence of results and proofs interspersed with definitions and comments.

\pause 

\item  At the other extreme, some authors state very few results formally.
A good example of the latter style is the classic book {\it\color{blue}The Algebraic Eigenvalue Problem} [296] by Wilkinson, in which only four titled theorems are given in 662 pages. As Boas [33] notes, ``A great deal can be accomplished with arguments that fall short of being formal proofs.''
  
\end{myitemize}

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\begin{frame}{ What is a Theorem? - 5  }

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\begin{myitemize}
\item  {\bf\color{red}A fifth kind of statement used in mathematical writing is a conjecture -- a statement that the author thinks may be true but has been unable to prove or disprove.}

\pause 

\item  The author will usually have some strong evidence for the veracity of the statement.
A famous example of a conjecture is the Goldbach conjecture (1742), which states that every even number greater
than 2 is the sum of two primes; this is still unproved.

\pause 

\item {\bf\color{red}A hypothesis is a statement that is taken as a basis for further reasoning, usually in a proof -- for example, an induction hypothesis.} 

\pause 

\item Hypotheses that stand on their own are uncommon; two examples are the Riemann hypothesis and the continuum hypothesis.

\end{myitemize}

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%\begin{frame}{ What is a Theorem? - 6  }
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%
%\begin{myitemize}
%
%\item One computer scientist (let us call him Alpha) joked in a talk
%``This is the Alpha and Beta conjecture. If it turns out to be false I would like it to be known as Beta's conjecture.''
%
%\item However, it is not necessarily a bad thing to make a conjecture that is later disproved: identifying the question that the conjecture aims to answer can be an important contribution.
%
%\item {\bf\color{red}A hypothesis is a statement that is taken as a basis for further reasoning, usually in a proof -- for example, an induction hypothesis. }
%
%\item Hypotheses that stand on their own are uncommon; two examples are the Riemann hypothesis and the continuum hypothesis.
%
%\end{myitemize}
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%\end{frame}%\vfill\hfill\thepage/\pageref{LastPage}\newpage
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\begin{frame}{ Proofs - 1  }

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\begin{myitemize}
\item  Readers are often not very interested in the details of a proof but want to know the outline and the key ideas.

\pause 

\item They hope to learn a technique or principle that can be applied in other situations.

\pause 

\item When readers do want to study the proof in detail they naturally want to understand it with the minimum of effort.

\pause 

\item {\bf\color{red}To help readers in both circumstances, it is important to emphasize the structure of a proof, the ease or difficulty of each step, and the key ideas that make it work. }

\end{myitemize}

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\begin{frame}{ Proofs - 2  }

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\begin{myitemize}
\item Here are some examples of the sorts of phrases that can be used (most of these are culled from proofs by Parlett in [217]).

\begin{center}
\fbox{
\begin{minipage}{10cm}
The aim/idea is to ...\\
Our first goal is to show that ... \\
Now for the harder part. ... \\
The trick of the proof is to find ... \\
... is the key relation. \\
The only, but crucial use of . .. is that ... \\
To obtain ... a little manipulation is needed. ... \\
The essential observation is that ... 
\end{minipage}}
\end{center}
  
\end{myitemize}

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\begin{frame}{ Proofs - 3  }

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\begin{myitemize}
\item  When you omit part of a proof it is best to indicate the nature and length of the omission, via phrases such as the following.

\begin{center}
\fbox{
\begin{minipage}{10cm}
It is easy/simple/straightforward to show that ... \\
Some tedious manipulation yields ... \\
An easy/obvious induction gives ... \\
After two applications of ... we find ... \\
An argument similar to the one used in ... shows that ... 
\end{minipage}}
\end{center}

\pause 

\item  The end of a proof is often marked by the {\bf\color{red}halmos symbol $\square$}.% (see the quote on page 24).

\pause 

\item Sometimes the abbreviation {\bf\color{red} QED} is used instead. \\
 (Latin: {\bf\color{red}quod erat demonstrandum} = which was to be demonstrated)

\end{myitemize}

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\begin{frame}{ Proofs - 4  }

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\begin{myitemize}
\item  You should also strive to keep the reader informed of where you are in the proof and what remains to be done.
Useful phrases include

\begin{center}
\fbox{
\begin{minipage}{10cm}
First, we establish that ... \\
Our task is now to ... \\
Our problem reduces to ... \\
It remains to show that ... \\
We are almost ready to invoke ... \\
We are now in a position to ... \\
Finally, we have to show that ... 
\end{minipage}}
\end{center}
\end{myitemize}

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%\begin{frame}{ Proofs - 5  }
%
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%
%\begin{myitemize}
%\item  The end of a proof is often marked by the {\bf\color{red}halmos symbol} $\square$ (see the quote on page 24).
%
%\item Sometimes the abbreviation QED (Latin: {\bf\color{red}quod erat demonstrandum} = which was to be demonstrated) is used instead.
%
%\item There is much more to be said about writing (and devising) proofs.
%References include Franklin and Daoud [85], Garnier and Taylor [101], Lamport [173], Leron [177] and Polya [228].
%
%\end{myitemize}
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\begin{frame}{ The Role of Examples - 1  }

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\begin{myitemize}
\item  {\bf\color{red}A pedagogical tactic that is applicable to all forms of technical writing (from teaching to research) is to discuss specific examples before the general case. }

\pause 

\item It is tempting, particularly for mathematicians, to adopt the opposite approach, but beginning with examples is often the more effective way to explain (see Boas's article [33] and the quote from it at the beginning of this chapter, a quote that itself illustrates this principle!).

\end{myitemize}

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\begin{myitemize}
\item  A good example of how to begin with a specific case is provided by Strang in Chapter 1 of {\it\color{blue} Introduction to Applied Mathematics} [262]:

\begin{center}
\fbox{\begin{minipage}{10cm}
The simplest model in applied mathematics is a system of linear
equations. It is also by far the most important, and we begin
this book with an extremely modest example:
\begin{eqnarray*}
\left\{\begin{array}{l}
2x_l + 4x_2 = 2,\\
4x_l + 11x_2 = 1.
\end{array}\right.
\end{eqnarray*}
\end{minipage}}
\end{center}

\pause 

\item After some further introductory remarks, Strang goes on to study in detail both this $2\times 2$ system and a particular $4\times 4$ system. General $n \times n$ matrices appear only several pages later.

  
\end{myitemize}

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\begin{frame}{ The Role of Examples - 3  }

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\begin{myitemize}
\item  Another example is provided by Watkins's {\it\color{blue} Fundamentals of Matrix Computations} [289]. 

\pause 

\item Whereas most linear algebra textbooks introduce Gaussian elimination for general matrices before discussing Cholesky factorization for symmetric positive definite matrices, Watkins reverses the order, giving the more specific but algorithmically more straightforward method first.

\pause 

\item  {\bf\color{red}An exercise in a textbook is a form of example.} I saw a telling criticism in one book review that complained ``The first exercise in the book was pointless, so why do the others?'' 

\end{myitemize}

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\begin{frame}{ The Role of Examples - 4  }

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\begin{myitemize}

\item {\bf\color{red}To avoid such criticism, it is important to choose exercises and examples that have a clear purpose and illustrate a point.} The first few exercises and examples should be among the best, to gain the reader's confidence. 

\pause 

\item The same reviewer complained of another book that ``it hides information in exercises and contains exercises that are too difficult.'' 

\pause 

\item Whether such criticism is valid depends on your opinion of what are the key issues to be transmitted to the reader and on the level of the readership. Again, it helps to bear such potential criticism in mind when you write.

\end{myitemize}

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\begin{frame}{ Definitions - 1 }

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\begin{myitemize}
\item  Three questions to be considered when formulating a definition are ``why?'', ``where?'' and ``how?'' 

\pause 

\item  {\bf\color{red}First, ask yourself why you are making a definition: is it really necessary?} Inappropriate definitions can complicate a presentation and too many can overwhelm a reader, so it is wise to imagine yourself being charged a large sum for each one. 

\pause 

\item  Instead of defining a square matrix $A$ to be {\it\color{blue}contractive} with respect to a norm $\mid\cdot\mid$ if $|A|<1$, which is not a standard definition, you could simply say ``A with $|A|<1$'' whenever necessary. This is easy to do if the property is needed on only a few occasions, and saves the reader having to remember what ``{\it\color{blue}contractive}'' means.
  
\end{myitemize}

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\begin{myitemize}
\item  For notation that is standard in a given subject area, judgement is needed to decide whether the definition should be given. 

\pause 

\item Potential confusion can often be avoided by using redundant words. For example, if $\rho (A)$ is not obviously the {\it\color{blue} spectral radius} of the matrix $A$ you can say ``the spectral radius $\rho(A)$''.

\pause 

\item  {\bf\color{red}The second question is ``where?'' The practice of giving a long sequence of definitions at the start of a work is not recommended.} Ideally, a definition should be given in the place where the term being defined is first used. 

\end{myitemize}

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\begin{myitemize}
\item If it is given much earlier, the reader will have to refer back, with a possible loss of concentration (or worse, interest). 
{\bf\color{red}Try to minimize the distance between a definition and its place of first use.}

\pause 

%\item  It is not uncommon for an author to forget to define a new term on its first occurrence. For example, Steenrod uses the term ``grasshopper reader'' on page 6 of his essay on mathematical writing [256], but does not define it until it occurs again on the next page.

\item {\bf\color{red}To reinforce notation that has not been used for a few pages you may be able to use redundancy.} For example, ``The  optimal step length $\alpha^*$ can be found as follows.'' This implicit redefinition either reminds readers what $\alpha^*$ is, or reassures them that they have remembered it correctly.

\pause 

\item  {\bf\color{red}Finally, how should a term be defined? There may be a unique definition or there may be several possibilities}.
% (a good example is the term $M$-matrix, which can be defined in at least fifty different ways [23]). 

\pause 

\item You should aim for a definition that is short, expressed in terms of a fundamental property or idea, and consistent with related definitions. 

\end{myitemize}

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\begin{frame}{ Definitions - 4 }

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\begin{myitemize}

\item As an example, the standard definition of a {\it\color{blue}normal matrix} is a matrix $A\in\mathbb{C}^{n\times n}$ for which $A^*A = AA^*$. % (where $*$ denotes the conjugate transpose). 
There are at least 70 different ways of characterizing normality [119], but none has the simplicity as this one.
% and ease of use of the condition $A^*A = AA^*$.

\pause 

\item  {\bf\color{red}By convention, {\it if} means {\it if and only if} in definitions}, so do not write 

\begin{center}
\fbox{\color{red}
\begin{minipage}{10cm}
``Definition. The graph $G$ is {\it\color{blue}connected} if and only if there is a path from every node in $G$ to every other node in G.'' 
\end{minipage}}
\end{center}

\pause 

\item  If you have not done so before, it is instructive to study the definitions in a good dictionary. They display many of the attributes of a good mathematical definition: they are concise, precise, consistent with other definitions, and easy to understand.

%\item Write ``The graph $G$ is connected if there is a path from every node in $G$ to every other node in $G$'' (and note that this definition can be rewritten to omit the symbol $G$). 

\end{myitemize}

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%
%\begin{myitemize}
%
%
%\item It is common practice to italicize the word that is being defined: ``A graph is {\it connected} if there is a path from every node to every other node.'' 
%
%\item This has the advantage of making it perfectly clear that a definition is being given, and not a result. This emphasis can also be imparted by writing ``A graph is defined to be connected if $\cdots$'' , or ``A graph is said to be connected if $\cdots$.''
%  
%\end{myitemize}
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\begin{myitemize}

\item Definitions of symbols are usually made with a simple equality, perhaps preceded by the word ``let'' if they are inline, as in ``let $q(x) = ax^2 +bx+c$.'' Various other notations have been devised to give emphasis to a definition.
%including

\pause 

\begin{center}
\fbox{
\begin{minipage}{10cm}
\begin{eqnarray*}
q(x) &:=& ax^2 + bx + c,\\
ax^2 + bx + c &=:& q(x),\\
q(x) &\overset{def}{=}& ax^2 + bx + c,\\
q(x) &\equiv& ax^2 +bx+c,\\
q(x) &\overset{\triangle}{=}& ax^2 +bx+c.
\end{eqnarray*}
\end{minipage}}
\end{center}

\pause 

\item If you use one of these special notations you must use it consistently. %, otherwise the reader may not know whether a straightforward equality is meant to be a definition.

\end{myitemize}

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\begin{frame}{ Notation - 1 }

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\begin{myitemize}
\item  The following extract is full of potentially confusing notation. 

\begin{center}\fbox{
\begin{minipage}{10cm}
Let $\hat{H_k} = Q_k^H\tilde{H_k}Q_k$, partition $X = [X_1, X_2]$ and let $\mathcal{X}= \textrm{range}(X_1)$.
Let $U^*$ denote the nearest orthonormal matrix to $X_1$ in the 2-norm.
\end{minipage}}
\end{center}

\pause 

    \item[1.]  The distinction between the hat and the tilde in $\hat{H_k}$ and $\tilde{H_k}$ is slight enough to make these symbols difficult to distinguish. 
    
\pause 

    \item[2.]  The symbols $\mathcal{X}$ and $X$ are also too similar for easy recognition. 
    
\pause 

    \item[3.] Given that $\mathcal{X}$ is used, it would be more consistent to give it a subscript 1. 
    
\pause 

    \item[4.] The name $H_k$ is unfortunate, because $H$ is being used to denote the {\it\color{blue}conjugate transpose}, and it might be necessary to refer to $\tilde{H_k}^H$. 
    %\item Since $A^*$ is a standard synonym for $A^H$, the use of a superscripted asterisk to denote optimality is confusing.
    
\end{myitemize}

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\begin{frame}{ Notation - 2 }

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\begin{myitemize}
%\item  %As this example shows, the choice of notation deserves careful thought.
\item  {\bf\color{red}Good notation strikes a balance among the possibly conflicting aims of being readable, natural, conventional, concise, logical and aesthetically pleasing.} As with definitions, the amount of notation should be minimized.

\pause 

%\item Although there are 26 letters in the alphabet and nearly as many again in the Greek alphabet, our choice diminishes rapidly when we consider existing connotations. 

\item Traditionally, $\epsilon$ and $\delta$ denote small quantities, $i$, $j$, $k$, $m$ and $n$ are integers (or $i$ or $j$ the imaginary unit), $\lambda$ is an eigenvalue and $\pi$ and $e$ are fundamental constants; $\pi$ is also used to denote a permutation. 

\pause 

\item {\bf\color{red}These conventions should be respected.} But by modifying and combining eligible letters we widen our choice. Thus $\gamma$ and $A$ yield, for example, $\hat{A}$, $\bar{A}$, $\tilde{A}$, $A'$, $\gamma_A$, $A_\gamma$, $\mathcal{A}$, $\mathbb{A}$.
 
\end{myitemize}

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%每页详细内容

\begin{myitemize}
\item {\bf\color{red}Particular areas of mathematics have their own notational conventions. }
For example, in numerical linear algebra lower case Greek letters represent scalars, lower case roman letters represent column vectors, and upper case Greek or roman letters represent matrices. This convention was introduced by Householder [143].

\pause 

\item  In his book on the symmetric eigenvalue problem [217], Parlett uses the symmetric letters $A$, $H$, $M$, $T$, $U$, $V$, $W$, $X$, $Y$ to denote symmetric matrices and the symmetric Greek letters $\Lambda$, $\Theta$, $\Phi$, $\Delta$ to denote diagonal matrices. 

%\item Actually, the roman letters printed above are not symmetric because they are slanted, but Parlett's book uses a sans serif mathematics font that yields the desired symmetry. Parlett uses this elegant, but restrictive, convention to good effect.

\end{myitemize}

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%每页详细内容

\begin{myitemize}
\item  We can sometimes simplify an expression by giving a meaning to extreme cases of notation. Consider the display
\begin{eqnarray*}
\beta_{ij} =\left\{
\begin{array}{ll}
0, & i>j \\
\frac{1}{u_j}, & i=j \\
\frac{1}{u_j}\prod\limits_{r=i}^{j-1}\left( \frac{-c_r}{u_r}\right), & i<j.
\end{array}\right.
\end{eqnarray*}

\pause 

\item There are really only two cases: $i > j$ and $i \le j$. This structure is reflected and the display made more compact if we define the empty product to be 1, and write
\begin{eqnarray*}
\beta_{ij} =\left\{
\begin{array}{ll}
0, & \textrm{ if } i>j \\
\frac{1}{u_j}\prod\limits_{r=i}^{j-1}\left( \frac{-c_r}{u_r}\right), &\textrm{ if } i\le j.
\end{array}\right.
\end{eqnarray*}

\end{myitemize}

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%每页详细内容

\begin{myitemize}
\item Here, I have put ``if'' before each condition, which is optional in this type of display. Incidentally, note that in a matrix product the order of evaluation needs to be specified: $\Pi_{i=1}^{n}A_i$ could mean $A_1A_2\cdots A_n$ or $A_nA_{n-1}\cdots A_1$.

\pause 

\item  {\bf\color{red}Notation also plays a higher level role in affecting the way a method or proof is presented.} For example, the $n\times n$ matrix multiplication $C = AB$ can be expressed in terms of scalars,
\begin{eqnarray*}
c_{ij} = \sum\limits_{k=1}^{n} a_{ik}b_{kj}, \quad 1\le i,j\le n,
\end{eqnarray*}
or at the matrix-vector level,
%\begin{eqnarray*}
$C = [Ab_1,Ab_2,\cdots,Ab_n]$, 
%\end{eqnarray*}
where $B = [b_1, b_2,\cdots, b_n]$ is a partition into columns. 
  
\end{myitemize}

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%每页详细内容

\begin{myitemize}
\item  {\bf\color{red}One of these two viewpoints may be superior, depending on the circumstances.} A deeper example is provided by the fast Fourier transform. 

\pause 

\item The discrete Fourier transform (DFT) is a product $y = F_nx$, where $F_n$ is the unitary Vandermonde matrix with $(r,s)$ element $\omega^{(r-1)(s-1)}$ $(1 < r,s < n)$, and $\omega = \exp( -2\pi i/n)$. The FFT is a way of forming this product in $O(n\log n)$ operations.

\pause 

\item It is traditionally expressed through equations such as the following (copied from a numerical methods text book):
\begin{eqnarray*}
\sum\limits_{j=0}^{n-1} e^{2\pi ijk/n} f_j =
\sum\limits_{j=0}^{n/2-1} e^{2\pi ikj/(n/2)} f_{2j} +
\omega^k\sum\limits_{j=0}^{n/2-1} e^{2\pi ikj/(n/2)} f_{2j+1}.
\end{eqnarray*}

\end{myitemize}

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%每页详细内容

\begin{myitemize}
\item  The language of matrix factorizations can be used to give a higher level description. If $n = 2m$, the matrix $F_n$ can be factorized as
\begin{eqnarray*}
F_n\Pi_n = \begin{bmatrix} I_m & \Omega_m \\ I_m & -\Omega_m  \end{bmatrix}
\begin{bmatrix} F_m & 0 \\ 0 & F_m  \end{bmatrix},
\end{eqnarray*}
where $\Pi_n$ is a permutation matrix and $\Omega_m = \text{diag}(1,\omega,\cdots,\omega^{m-1})$. 

\pause 

\item This factorization shows that an $n$-point DFT can be computed from two $n/2$-point transforms, and this reduction is the gist of the radix-2 FFT. 

\pause 

\item The book {\it\color{blue} Computational Frameworks for the Fast Fourier Transform} by Van Loan [284], from which this factorization is taken, shows how, {\bf\color{red}by using matrix notation, the many variants of the FFT can be unified and made easier to understand}.

\end{myitemize}

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%
%\begin{myitemize}
%\item  An extended example of how notation can be improved is given by Gillman in the appendix titled ``{\it The Use of Symbols: A Case Study}'' of {\it Writing Mathematics Well} [104]. 
%
%\item {\bf\color{red}Gillman takes the proof of a theorem by Sierpinski (1933) and shows how simplifying the notation leads to a better proof.} Knuth set his students the task of simplifying Gillman's version even further, and four solutions are given in [164, 321].
%
%\end{myitemize}
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%每页详细内容

\begin{myitemize}
\item {\bf\color{red}Mathematicians are always searching for better notation.} Knuth [163] describes two notations that he and his students have been using for many years and that he thinks deserve widespread adoption. 

\pause 

\item One is notation for the Stirling numbers. The other is the notation $[S]$, where $S$ is any true-or-false statement. The definition is
\begin{eqnarray*}
[S] =\left\{
\begin{array}{ll}
1, & \text{ if } S \text{ is true},\\
0, & \text{ if } S \text{ is false}.
\end{array}
\right.
\end{eqnarray*}

\pause 

\item For example, the Kronecker delta can be expressed as $\delta_{ij} = [i = j]$. 

%{\bf\color{red}The square bracket notation will seem natural to those who program; indeed, Knuth adapted it from a similar notation in the 1962 book by Iverson that led to the programming language APL [144, p.11]. }

%\item The square bracket notation is used in the textbook {\it Concrete Mathematics} [116]; that book and Knuth's paper give a convincing demonstration of the usefulness of the notation.

\end{myitemize}

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%\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%%每页详细内容
%
%\begin{myitemize}
%\item  Halmos has these words to say about two of his contributions to mathematical notation [127]:
%\begin{center}
%\fbox{
%\begin{minipage}{12cm}
%My most nearly immortal contributions to mathematics are an abbreviation and a typographical symbol. {\bf\color{red}I invented ``iff,'' for ``if and only if'' - but I could never believe that I was really its first inventor.} ... The symbol is definitely not my invention - it appeared in popular magazines (not mathematical ones) before I adopted it, but, once again, I seem to have introduced it into mathematics. {\bf\color{red}It is the symbol that sometimes looks like $\square$, and
%is used to indicate an end, usually the end of a proof.} It is most frequently called the ``tombstone,'' but at least one generous author referred to it as the ``halmos''.
%\end{minipage}}
%\end{center}
%  
%\end{myitemize}
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%每页详细内容

\begin{myitemize}
\item  Table 3.1 shows the date of first use in print of some standard symbols; some of them are not as old as you might expect. Not all these notations met with approval when they were introduced. 

\pause 

\item In 1842 Augustus de Morgan complained (quoted by Cajori [49, p. 328 (Vol. II)]):
\begin{center}
\fbox{
\begin{minipage}{12cm}
Among the worst of barbarisms is that of introducing symbols which are quite new in mathematical, but perfectly understood in common language. Writers have borrowed from the Germans the abbreviation $n!$ to signify $1\cdot 2\cdot 3\cdot \cdots \cdot (n-1)\cdot n$, which gives their pages the appearance of expressing surprise and admiration that 2, 3, 4, etc., should be found in mathematical results.
\end{minipage}}
\end{center}
  
\end{myitemize}

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%每页详细内容

%\begin{center}
\begin{table}[ht!]\centering
\caption{First use in print of some symbols}
\begin{tabular}{|l|l|l|}\hline
Symbol & Name & Year of publication \\ \hline
$\infty$ & infinity & 1655 (Wallis) \\
$\pi$  & pi $(3.14159...)$ & 1706 (Jones) \\
$e$  & $e(2.71828...)$ & 1736 (Euler) \\
$i$  & imaginary unit $(\sqrt{-1})$ & 1794 (Euler) \\
$\equiv$  & congruence & 1801 (Gauss) \\
$n!$  & factorial & 1808 (Kramp) \\
$\sum$  & summation & 1820 (Fourier) \\
$\binom{n}{k}$  & binomial coefficient & 1826 (von Ettinghausen) \\
\hline
\end{tabular}
\end{table}
%\end{center}


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%每页详细内容

%\begin{center}
\begin{table}[ht!]\centering
\caption{First use in print of some symbols (continued)}
\begin{tabular}{|l|l|l|}\hline
Symbol & Name & Year of publication \\ \hline
$\Pi$  & product & 1829 (Jacobi) \\
$\nabla$  & nabla & 1853 (Hamilton) \\
$\delta_{ij}$  & Kronecker delta & 1868 (Kronecker) \\
$|z|$  & absolute value & 1876 (Weierstrass) \\
$O(f(n))$  & big oh & 1894 (Bachmann) \\
$\llcorner x \lrcorner$  & floor & 1962 (Iverson) \\
$\ulcorner x \urcorner$  & ceil & 1962 (Iverson) \\
\hline
\end{tabular}
\end{table}
%\end{center}

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\begin{frame}{ Words versus Symbols - 1 }

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%每页详细内容

\begin{myitemize}
\item  Mathematicians are supposed to like numbers and symbols, but I think many of us prefer words. 

\pause 

\item  If we had to choose between reading a paper dominated by symbols and one dominated by words then, all other things being equal, most of us would choose the wordy paper, because we would expect it to be easier to understand. 

\pause 

\item  {\bf\color{red}One of the decisions constantly facing the mathematical writer is how to express ideas: in symbols, in words, or both. }

\item  I suggest some guidelines.

\end{myitemize}

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%每页详细内容


\begin{myenumerate}

\item {\color{red}Use symbols if the idea would be too cumbersome to express in words, or if it is important to make a precise mathematical statement.}

\pause 

\item {\color{red}Use words as long as they do not take up much more space than the corresponding symbols.}

\pause 

\item {\color{red}Explain in words what the symbols mean if you think the reader might have difficulty grasping the meaning or essential feature.}

\end{myenumerate}


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\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{myitemize}
\item  (1) Consider the example.

\begin{center}\fbox{
\begin{minipage}{10cm}
Define $C\in\mathbb{R}^{n\times n}$ by the property that $vec(C)$ is the eigenvector corresponding to the smallest eigenvalue in magnitude of $A$, where the $vec$ operator stacks the columns of a matrix into one long vector.
\end{minipage}}
\end{center}

\pause 

\item 
{\bf\color{red}To make this definition using equations takes much more space,} and is not worthwhile unless the notation that needs to be introduced (in this case, a name for $\text{min}\{ | \lambda |: \lambda \text{ is an eigenvalue of } A \} $ ) is used elsewhere. 

\pause 

\item 
A possible objection to the above wordy definition of {\it\color{blue}vec} is that it does not specify in which order the columns are stacked, but that can be overcome by appending ``taking the columns in order from first to last''. 
  
\end{myitemize}

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\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{myitemize}

\item  (2) Compare two sentences.

\begin{center}\fbox{
\begin{minipage}{10cm}
Since $|g'(0)|>1$, zero is a repelling fixed point, so $x_k$ does not tend to zero as $k\to\infty$.
\end{minipage}}
\end{center}

\begin{center}\fbox{
\begin{minipage}{10cm}
Since $|g'(0)|>1$, 0 is a repelling fixed point, so $x_k\nrightarrow 0$ as $k\to\infty$.
\end{minipage}}
\end{center}

\pause 

\item 
This sentence is only slightly shorter than the original and is harder to read -- the symbols are beginning to intrude on the grammatical structure of the sentence.

\end{myitemize}

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\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{myitemize}

\item  (3) Compare two sentences.

\begin{center}\fbox{
\begin{minipage}{10cm}
If $B\in\mathbb{R}^{n\times n}$ has a unique eigenvalue $\lambda$ of largest modulus then $B^k\approx \lambda^kxy^T$, where $Bx = \lambda x$ and $y^T x = \lambda y^T$ with $y^T x = 1$.
\end{minipage}}
\end{center}

\begin{center}\fbox{
\begin{minipage}{10cm}
If $B\in\mathbb{R}^{n\times n}$ has a unique eigenvalue $\lambda$ of largest modulus then $B^k\approx \lambda^kxy^T$, where $x$ and $y$ are a right and left eigenvector corresponding to $\lambda$, respectively, and $y^T x = 1$. 
\end{minipage}}
\end{center}

\pause 

\item The second sentence is cumbersome.

\end{myitemize}

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\begin{frame}{ Words versus Symbols - 6 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{myitemize}

\item (4) Consider the example.

\begin{center}\fbox{
\begin{minipage}{12cm}
 Under these conditions the perturbed least squares solution $x + \Delta x$ can be shown to satisfy
\begin{eqnarray*}
\frac{\Vert \Delta x\Vert_2}{\Vert x\Vert_2} \le 
\epsilon\kappa_2(A) \left( 1+ \frac{\Vert b\Vert_2}{\Vert A\Vert_2\Vert x\Vert_2  } \right)+
\epsilon\kappa_2(A)^2 \frac{\Vert r\Vert_2}{\Vert A\Vert_2\Vert x\Vert_2  } + O(\epsilon^2).
\end{eqnarray*}
Thus the sensitivity of $x$ is measured by $\kappa_2(A)$ if the residual $r$ is zero or small, and otherwise by $\kappa_2(A)^2$.
\end{minipage}}
\end{center}

\pause 

\item Here, we have a complicated bound that demands an explanation in words, lest the reader overlook the significant role played by the residual $r$.

\end{myitemize}

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\begin{frame}{ Words versus Symbols - 7 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{myitemize}

\item  (5) Compare two sentences.
\begin{center}\fbox{
\begin{minipage}{10cm}
If $y_1,y_2,\cdots, y_n$ are all $\neq 1$ then $g(y_1,y_2,\cdots,y_n)>0$.
\end{minipage}}
\end{center}

\begin{center}\fbox{
\begin{minipage}{10cm}
If $y_i\neq 1$ for $i = 1,2,\cdots,n$, then $g(y_l, y_2,\cdots, y_n)>0$.
If none of the $y_i (i = 1,2,..., n)$ equals 1, then $g(y_l, y_2,\cdots, y_n)>0$.
\end{minipage}}
\end{center}

\pause 

\item  In the first sentence ``all $\neq 1$'' is a clumsy juxtaposition of word and equation and most writers would express the statement differently. 

\pause 

\item If the condition were ``$\neq 0$'' instead of ``$\neq 1$'', then it could simply be replaced by the word ``nonzero''. 

\end{myitemize}

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\begin{frame}{ Words versus Symbols - 8 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{myitemize}

%\item {\bf\color{red}In cases such as this, the choice between words and symbols in the text (as opposed to in displayed equations) is a matter of taste; good taste is acquired by reading a lot of well-written mathematics.}

\item The symbols {\bf\color{red}$\forall$} and {\bf\color{red}$\exists$} are widely used in handwritten notes and are an intrinsic part of the language in logic. But generally, in equations that are in-line, they are better replaced by the equivalent words `` {\bf\color{red}for all}'' and ``{\bf\color{red}there exists}''. 

\pause 

\item  In displayed equations either the symbol or the phrase is acceptable, though I usually prefer the phrase. Compare
\begin{eqnarray*}
\sigma(G(t)) = \exp(t\sigma(A))\,\,\, \textrm{ for all } t > 0
\end{eqnarray*}
with
\begin{eqnarray*}
\sigma(G(t)) = \exp(t\sigma(A))\,\,\, \forall t>0.
\end{eqnarray*}
  
\end{myitemize}

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%\begin{frame}{ Words versus Symbols - 9 }
%
%\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%%每页详细内容
%
%\begin{myitemize}
%\item Similar comments apply to the symbols $\Rightarrow$ ( {\bf\color{red}implies}) and $\Leftrightarrow$ ( {\bf\color{red}if and only if}), though these symbols are more common in displayed formulas.
%
%\item Of course, for some standard phrases that appear in displayed formulas, there is no equivalent symbol:
%\begin{center}\fbox{
%\begin{minipage}{10cm}
%\begin{eqnarray*}
%\textrm{minimize } c^Tx -\mu\sum\limits_{i=1}^{n} \ln x_i\,\,\, \textrm{ subject to } Ax =b, \\
%\underline{\lim} \frac{1}{n}\log D_j^n <0\,\, \textrm{ almost surely}, \\
%Z^T y = \Vert z \Vert_D \Vert y\Vert = 1, \textrm{ where } \Vert z\Vert_D = 
%\underset{v\neq 0}{\max}\frac{|z^Tv|}{\Vert v\Vert}.
%\end{eqnarray*}
%\end{minipage}}
%\end{center}
%
%\end{myitemize}
%
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\begin{frame}{ Displaying Equations - 1 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{myitemize}

\item   {\bf\color{red}An equation is displayed when it needs to be numbered}, when it would be hard to read if placed in-line, or when it merits special attention, perhaps because it contains the first occurrence of an important variable.
The following extract gives an illustration of what and what not to display.

\end{myitemize}

\begin{center}
\fbox{\small 
\begin{minipage}{12cm}
Because $\delta(\bar{x},\mu)$ is the smallest value of $\Vert \bar{X}z/\mu -e\Vert$ for all vectors $y$ and $z$ satisfying $A^T y + z = c$, we have
$$\delta(\bar{x},\mu) < \Vert \frac{1}{\mu}\bar{X}z - e\Vert.$$
Using the relations $z = \mu X^{-1}s$ and $\bar{x}_i = 2x_i - x_is_i$ gives 
$$ \frac{1}{\mu}\bar{X}z = \bar{X} X^{-1} s = (2X - XS)X^{-1} s = 2s - S^2e.$$
\end{minipage}}
\end{center}


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\begin{frame}{ Displaying Equations - 2 }

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%每页详细内容

\begin{center}\fbox{
\begin{minipage}{12cm}
Therefore, $\delta(\bar{x},\mu) < \Vert 2s - S^2e - e\Vert$, which means that
{\footnotesize $$\delta(\bar{x},\mu)^2 
\le \sum\limits_{i=1}^{n} (2s_i - s_i^2 - 1)^2 
\sum\limits_{i=1}^{n} (s_i - 1)^4 
\le \left(\sum\limits_{i=1}^{n} (s_i - 1)^2 \right)^2
=\delta(x,\mu)^4.$$}
The condition $\delta(x,\mu)<1$ thus ensures that the Newton iterates $\bar{x}$ converge quadratically.
\end{minipage}}
\end{center}

\begin{myitemize}
\item {\bf\color{red}The second and third displayed equations are too complicated to put in-line.} The first $\delta(\bar{x}, \mu)$ inequality is displayed because it is used in conjunction with the second display and it is helpful to the reader to display both these steps of the argument. The consequent inequality %$\delta(\bar{x},\mu) < \Vert 2s-S^2e - e\Vert$ 
fits nicely in-line, and since it is used immediately it is not necessary to display it.
\end{myitemize}

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\begin{frame}{ Displaying Equations - 3 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{myitemize}

\item  {\bf\color{red}When a displayed formula is too long to fit on one line it should be broken before a binary operation.} Example:
\begin{center}\fbox{
\begin{minipage}{12cm}
\begin{eqnarray*}
|e_{m+1}| &<&  G^{m+1} e_0 + c_n u(1 +\theta_x)\{c(A)|(I-G)^D M^{-1}| \\
 && + (m+1)|(I-E)M^{-1}|\} (|M| + |N|)|x|.
\end{eqnarray*}
\end{minipage}}
\end{center}

\pause 

\item  The indentation on the second line should take the continuation expression past the beginning of the left operand of the binary operation at which the break occurred, though, as this example illustrates, this is not always
possible for long expressions. 

\pause 

\item  A formula in the text should be broken after a relation symbol or binary operation symbol, not before.

\end{myitemize}

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\begin{frame}{ Parallelism - 1 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{myitemize}

\item  {\bf\color{red}Parallelism should be used, where appropriate, to aid readability and understanding.} Consider this extract:
\begin{center}\fbox{
\begin{minipage}{12cm}
The Cayley transform is defined by $C = (A - \theta_1I)^{-1}(A - \theta_2I)$.
If $\lambda$ is an eigenvalue of $A$ then
$$(\lambda - \theta_2)(\lambda - \theta_1)^{-1}$$
is an eigenvalue of $C$.
\end{minipage}}
\end{center}

\pause 

\item  The factors in the eigenvalue expression are presented in the reverse order to the factors in the expression for $C$. This may confuse the reader, who might, at first, think there is an error. The two expressions should be ordered in the same way.

\end{myitemize}

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\begin{frame}{ Parallelism - 2 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{myitemize}

\item  {\bf\color{red}Parallelism works at many levels, from equations and sentences to theorem statements and section headings.} It should be borne in mind throughout the writing process. 

\pause 

\item If one theorem is very similar to another, the statements should reflect that -- the wording should not be changed just for the sake of variety (see {\it\color{blue}elegant variation}, \S 4.15). However, it is perfectly acceptable to economize on words by saying, in Theorem 2 (say) ``Under the conditions of Theorem 1''.

\end{myitemize}

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\begin{frame}{ Parallelism - 3 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{myitemize}
\item  For a more subtle example, consider the sentence
\begin{center}\fbox{
\begin{minipage}{10cm}
It is easy to see that $f(x, y) > 0$ for $x > y$.
\end{minipage}}
\end{center}

\item  
In words, this sentence is read as ``It is easy to see that $f(x, y)$ is greater than zero for $x$ greater than $y$.'' 

\pause 

\item The first $>$ translates to ``is greater than'' and the second to ``greater than'', so there is a lack of parallelism, which the reader may find disturbing. A simple cure is to rewrite the sentence:

\begin{center}\fbox{
\begin{minipage}{10cm}
It is easy to see that $f(x, y) > 0$ when $x > y$.\\
It is easy to see that if $x > y$ then $f(x, y) > 0$.
\end{minipage}}
\end{center}


\end{myitemize}

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\begin{frame}{ Dos and Don'ts - 1: Punctuating Expressions }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{myitemize}
\item {\bf\color{red}Mathematical expressions are part of the sentence and so should be punctuated.} In the following display, all the punctuation marks are necessary. (The second displayed equation might be better moved in-line.)

\begin{center}\fbox{
\begin{minipage}{12cm}
The three most commonly used matrix norms in numerical analysis are particular cases of the Holder $p$-norm
\begin{eqnarray*}
\Vert A\Vert_p = \underset{x\neq 0}{\max} \frac{\Vert Ax\Vert_p}{\Vert x\Vert_p}, \,\, A\in\mathbb{R}^{n\times n},
\end{eqnarray*}
where $p > 1$ and
\begin{eqnarray*}
\Vert x\Vert_p  = \left( \sum\limits_{i=1}^{n} |x_i|^p \right)^{1/p}. 
\end{eqnarray*}
\end{minipage}}
\end{center}

\end{myitemize}

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\begin{frame}{ Dos and Don'ts - 2: Otiose Symbols }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{myitemize}

\item  {\bf\color{red}Do not use mathematical symbols unless they serve a purpose.} 
    \begin{myitemize}
    \item  In the sentence
    \begin{center}\fbox{
    \begin{minipage}{10cm}
    ``A symmetric positive definite matrix $A$ has real eigenvalues'', 
    \end{minipage}}
    \end{center}
      there is no need to name the matrix unless the name is used in a following sentence. 
    
\pause 

\item  Similarly, in the sentence 
    \begin{center}\fbox{
    \begin{minipage}{10cm}
    ``This algorithm has $t = \log_2 n$ stages'', 
    \end{minipage}}
    \end{center}
the ``$t = $'' can be omitted unless $t$ is defined in this sentence and used immediately. 
    
\pause 

    \item  Watch out for unnecessary parentheses, as in the phrase ``the matrix $(A - \lambda I)$ is singular.''
    \end{myitemize}

\end{myitemize}

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\begin{frame}{ Dos and Don'ts - 3: Placement of Symbols }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{myitemize}

\item  {\bf\color{red}Avoid starting a sentence with a mathematical expression, particularly if a previous sentence ended with one, otherwise the reader may have difficulty parsing the sentence. }

\pause 

\item  For example, ``$A$ is an ill-conditioned matrix'' (possible confusion with the word ``$A$'') can be changed to ``The matrix $A$ is ill-conditioned.''

\end{myitemize}

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\begin{frame}{ Dos and Don'ts - 4  }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{myitemize}

\item  {\bf\color{red}Separate mathematical symbols by punctuation marks or words, if possible, for the same reason.}

\begin{center}\fbox{
\begin{minipage}{12cm}
    \begin{itemize}
    \item Bad: If $x > 1$ $f(x) < 0$.
    \item Fair: If $x > 1$, $f( x) < 0$.
    \item Good: If $x > 1$ then $f(x) < 0$.
    \end{itemize}
\end{minipage}}
\end{center}

\pause 

\begin{center}\fbox{
\begin{minipage}{12cm}
    \begin{itemize}
    \item Bad: Since $p^{-l} + q^{-l} = 1$, $\Vert\cdot\Vert_p$ and $\Vert\cdot\Vert_q$ are dual norms.
    \item Good: Since $p^{-l} + q^{-l} = 1$, the norms $\Vert\cdot\Vert_p$ and $\Vert\cdot\Vert_q$ are dual.
    \end{itemize}
\end{minipage}}
\end{center}

\end{myitemize}

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\begin{frame}{ Dos and Don'ts - 5: Otiose Symbols }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{center}\fbox{
\begin{minipage}{12cm}
    \begin{itemize}
    \item Bad: It suffices to show that $\Vert H\Vert_p = n^{1/p}, 1\le p\le 2$.
    \item Good: It suffices to show that $\Vert H\Vert_p = n^{1/p}$ for $1\le p\le 2$.
    \item Good: It suffices to show that $\Vert H\Vert_p = n^{1/p} (1\le p\le 2)$.
    \end{itemize}
\end{minipage}}
\end{center}

\pause 

\begin{center}\fbox{
\begin{minipage}{12cm}
    \begin{itemize}
    \item Bad: For $n = r$ (2.2) holds with $\delta_r = 0$.
    \item Good: For $n = r$, (2.2) holds with $\delta_r = 0$.
    \item Good: For $n = r$, inequality (2.2) holds with $\delta_r = 0$.
    \end{itemize}
\end{minipage}}
\end{center}

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\begin{frame}{ Dos and Don'ts - 6: ``The'' or ``A'' }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{myitemize}

\item {\bf\color{red}In mathematical writing the use of the article ``the'' can be inappropriate when the object to which it refers is (potentially) not unique or does not exist.} Rewording, or changing the article to ``a'', usually solves the problem.

\end{myitemize}

\begin{center}\fbox{
\begin{minipage}{12cm}
\begin{itemize}
\item Bad: Let the Schur decomposition of $A$ be $QTQ^*$.
\item Good: Let a Schur decomposition of $A$ be $QTQ^*$.
\item Good: Let $A$ have the Schur decomposition $QTQ^*$.
\end{itemize}
\end{minipage}}
\end{center}

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%\begin{frame}{ Dos and Don'ts - 7: ``The'' or ``A'' }
%
%\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%%每页详细内容

\pause 

\begin{center}\fbox{
\begin{minipage}{12cm}
\begin{itemize}
\item Bad: Under what conditions does the iteration converge to the solution of $f(x) = 0$?
\item Good: Under what conditions does the iteration converge to a solution of $f(x) = 0$?
\end{itemize}
\end{minipage}}
\end{center}

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\begin{frame}{ Dos and Don'ts - 7: Notational Synonyms }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{myitemize}

\item {\bf\color{red}Sometimes you have a choice of notational synonyms, one of which is preferable.} In the following examples, the first of each pair is, to me, the more aesthetically pleasing or easier to read.% (a capital letter denotes a matrix).

\begin{center}\fbox{
\begin{minipage}{12cm}
\begin{eqnarray*}
%\left( \sum\limits_{i,j} (a_{ij}-b_{ij})^2 \right)^{1/2}, && \sqrt{\sum\limits_{i,j} (a_{i,j}-b_{i,j})^2}, \\
\exp(2\pi i (x^2+y^2)^{-1/2}), && \frac{2\pi i}{e^{\sqrt{x^2+y^2}}}, \\
(1-n\epsilon)^{-1}|L||U|, && \frac{|L||U|}{1-n\epsilon}, \\
X_{k+1}=\frac{1}{2}X_k(3I-X_k^2), && X_{k+1}=\frac{X_k}{2}[3I-X_k^2],\\
\min\{\epsilon: |b-Ay|\le\epsilon|A||y|\}, && \min\{\epsilon \mid |b-Ay|\le\epsilon|A||y|\}.
\end{eqnarray*}
\end{minipage}}
\end{center}

\end{myitemize}

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\begin{frame}{ Dos and Don'ts - 8: Notational Synonyms }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{myitemize}

\item {\bf\color{red}In the next two examples, the first form is preferable because it saves space without a loss of readability.} Of course, the $\textrm{diag}(\cdot)$ notation should be defined if it is not regarded as standard.

\begin{center}\fbox{\footnotesize
\begin{minipage}{12cm}
\begin{eqnarray*}
x=\begin{bmatrix} x_1,x_2,\cdots,x_n \end{bmatrix}^T, && 
x=\begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}, \\
\Lambda = \textrm{diag}(\lambda_i), && 
\Lambda = \begin{bmatrix} \lambda_1 &&& \\ &\lambda_2&& \\ &&\ddots& \\ &&&\lambda_n \end{bmatrix}.
\end{eqnarray*}
\end{minipage}}
\end{center}

\end{myitemize}

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\begin{frame}{ Dos and Don'ts - 9: Referencing Equations }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{myitemize}

\item  {\bf\color{red}When you reference an earlier equation it helps the reader if you add a word or phrase describing the nature of that equation.} The aim is to save the reader the trouble of turning back to look at the earlier equation. 

\pause 

\item  For example, ``From the definition (6.2) of dual norm'' is more helpful than ``From (6.2)''; and ``Combining the recurrence (3.14) with inequality (2.9)'' is more helpful than ``Combining (3.14) and (2.9)''. {\bf\color{red}Mermin [200] calls this advice the ``Good Samaritan Rule''. }

\pause 

\item  As in these examples, the word added should be something more informative than just ``equation'' (or the ugly abbreviation ``Eq.''), and inequalities, implications and lone expressions should not be referred to as equations.

\end{myitemize}

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\begin{frame}{ Miscellaneous - 1 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{myitemize}

\item  When working with complex numbers it is best not to use ``$i$'' as a counting index, to avoid confusion with the imaginary unit. More generally, do not use a letter as a dummy variable if it is already being used for another purpose.

\pause 

\item  Note the difference between the Greek letter epsilon, $\epsilon$, and the ``belongs to'' symbol $\in$, as in $\Vert x\Vert \le\epsilon$ and $x\in\mathbb{R}^n$. Another version of the Greek epsilon is $\varepsilon$. Note the distinction between the Greek letter $\pi$ and the product symbol $\Pi$.

\pause 

\item  {\bf\color{red}By convention, standard mathematical functions such as $\sin$, $\cos$, $\arctan$, $\max$, $\gcd$, $\textrm{trace}$ and $\lim$ are set in roman type, as are multiple-letter variable names.} It is a common mistake to set these in italic type, which is ambiguous. For example, is $tan x$ the product of four scalars or the tangent of $x$?

\end{myitemize}

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\begin{frame}{ Miscellaneous - 2 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{myitemize}

\item  {\bf\color{red}In bracketing multilayered expressions you have a choice of brackets for the layers and a choice of sizes}, for example $\{[ ( \{[ ($, this ordering being the one recommended by {\it\color{blue} The Chicago Manual of Style} [58]. Most authors try to avoid mixing different brackets in the same expression, as it leads to a rather muddled appearance.

\pause 

\item  Write ``the $k$th term'' not ``the $k^{\textrm{th}}$ term'', ``the $k$'th term'' or "the $k$-th term." (It is interesting to note that nth is a genuine word that can be found in most dictionaries.)

\pause 

\item  A slashed exponent, as in $y^{1/2}$, is generally preferable to a stacked one, as in $y^{\frac{1}{2}}$.

\end{myitemize}

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\begin{frame}{ Miscellaneous - 3 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{myitemize}

\item  {\bf\color{red}The standard way to express that $i$ is to take the values $1$ to $n$ in steps of $1$ is to write 
\[ i=1,\cdots,n \,\,\textrm{ or }\,\, i=1,2,\cdots,n,\]
where all the commas are required. }

\pause 

\item  An alternative notation originating in programming languages such as Fortran 90 and MATLAB is $i = 1:n$.

\pause 

\item  For counting down we can write $i = n, n - 1,\cdots,1$ or $i = n:-1:1$, where the middle integer denotes the increment. This notation is particularly convenient when extended to describe submatrices: $A( i:j, p:q)$ denotes the submatrix formed from the intersection of rows $i$ to $j$ and columns $p$ to $q$ of the matrix $A$.

\end{myitemize}

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\begin{frame}{ Miscellaneous - 4 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{myitemize}

\item  Avoid (or rewrite) tall in-line expressions, such as $\begin{bmatrix}g_1 \\ g_2\end{bmatrix}$, which can disrupt the line spacing.

\pause 

\item  There are two different kinds of ellipsis: vertically centred $(\cdots)$ and ``ground level'' or ``baseline'' (...). Generally, the former is used between operators such as $+$, $=$, and $<$, and the latter is used between a list of 
symbols or to indicate a product. Examples:
\[ x_1+x_2+\cdots + x_n,\,\, \sigma_1\ge\sigma_2\ge\cdots\ge\sigma_n,\,\, \lambda_1\lambda_2...\lambda_n,\,\, A_1A_2...A_n. \]

\pause 

\item  {\bf\color{red}An operator or comma should be symmetrically placed around the ellipsis; thus $x_1 + x_2 +\cdots x_n$ and $\lambda_1,  \lambda_2,\cdots \lambda_n$ are incorrect.}

\end{myitemize}

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\begin{frame}{ Miscellaneous - 5 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{myitemize}

\item  {\bf\color{red}When an ellipsis falls at the end of a sentence there is the question of how the full stop (or period) is treated.} Recommendations vary. {\it\color{blue}The Chicago Manual of Style} suggests typing the full stop before the three ellipsis points
(so that there is no space between the first of the four dots and the preceding character). 

\pause 

\item  When the ellipsis is part of a mathematical formula it seems natural to put it before the full stop, but the two possibilities may be visually indistinguishable, as in the sentence 

\begin{center}\fbox{
\begin{minipage}{10cm}
The Mandelbrot set is defined in terms of the iteration 
$$z_{k+1} = z_k^2 + c, k = 0,1,2,\cdots.$$
\end{minipage}}
\end{center}

\end{myitemize}

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\begin{frame}{ Miscellaneous - 6 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{myitemize}

\item  A vertically centred dot is useful for denoting multiplication in expressions where terms need to be separated for clarity:
\begin{eqnarray*}
16046641 &=& 13\cdot 37\cdot 73\cdot 457, \\
\textrm{cond}(A,x) &=& \frac{\Vert \vert I-A^+A\vert \cdot \vert A^T\vert \cdot \vert{A^+}^Tx\vert \Vert}{\Vert x\Vert}.
\end{eqnarray*}

\pause 

\item  {\bf\color{red}Care is needed to avoid ambiguity in slashed fractions.} For example, the expression $-(b - a)^3/12f''(\eta)$ is better written as $-((b-a)^3/12)f''(\eta)$ or $-f''(\eta)(b-a)^3/12$.

\end{myitemize}

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\begin{frame}{ Glossary for Mathematical Writing }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{center}
\begin{tabular}{|p{7cm}|p{6cm}|} \hline
Without loss of generality & I have done an easy special case \\ \hline 
By a straightforward computation & I lost my notes  \\ \hline 
The details are left to the reader & I can't do it  \\ \hline 
The following alternative proof of X's result may be of interest & I cannot understand X  \\ \hline 
It will be observed that & I hope you hadn't noticed that  \\ \hline 
Correct to within an order of magnitude & wrong  \\ \hline 
\end{tabular}
\end{center}

-- Adapted from: 
H. Petard, {\it\color{blue}A Brief Dictionary of Phrases Used In Mathematical Writing}, American Mathematical Monthly.

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\end{document}




\begin{comment}


%\begin{enumerate}
%\item  Without loss of generality = I have done an easy special case.
%\item  By a straightforward computation = I lost my notes.
%\item  The details are left to the reader = I can't do it.
%\item  The following alternative proof of X's result may be of interest = I cannot understand X.
%\item  It will be observed that = I hope you hadn't noticed that.
%\item  Correct to within an order of magnitude = wrong.
%\end{enumerate}

\subsubsection{A Brief Dictionary of Phrases Used In Mathematical Writing}

Since authors seldom, if ever, say what they mean, the following glossary is offered to neophytes in mathematical research to help them understand the language that surrounds the formulas. Since mathematical writing, like mathematics, involves many undefined concepts, it seems best to illustrate the usage by interpretation of examples rather than to attempt definition.

\begin{enumerate}
\item 
ANALOGUE - This is an analogue of: I have to have some excuse for publishing it.
\item 
APPLICATIONS - This is of interest in applications: I have to have some excuse for publishing it.
\item 
COMPLETE - The proof is now complete: I can't finish it.
\item 
DETAILS - I cannot follow the details of X's proof: It's wrong.
\item 
DIFFICULT - This problem is difficult: I don't know the answer. (Cf. Trivial)
\item 
GENERALITY - Without loss of generality: I have done an easy special case.
\item 
IDEAS - To fix the ideas: To consider the only case I can do.
\item 
INGENIOUS - X's proof is ingenious: I understand it.
\item 
INTEREST - It may be of interest: I have to have some excuse for publishing it.
\item 
INTERESTING - X's proof is interesting: I don't understand it.
\item 
KNOWN - This is a known result but I reproduce the proof for the convenience of the reader: My paper isn't long enough.
\item 
LANGUAGE - PAR ABUS DE Language: In the terminology used by other authors. (Cf. Notation)
\item 
NATURAL - It is natural to begin with the following considerations: We have to start somewhere.
\item 
NEW - This was proved by X but the following new proof may present points of interest: I can't understand X.
\item 
NOTATION - To simplify the notation: It is too much trouble to change now.
\item 
OBSERVED - It will be observed that: I hope you have not noticed that.
\item 
READER - The details may be left to the reader: I can't do it.
\item 
REFEREE - I wish to thank the referee for the suggestions: I loused it up.
\item 
STRAIGHTFORWARD - By a straightforward computation: I lost my notes.
\item 
TRIVIAL - This problem is trivial: I know the answer. (Cf. difficult)
\item 
WELL-KNOWN - This result is well-known: I can't find the reference.

\end{enumerate}

EXERCISES FOR THE STUDENT - Interpret the following:

\begin{enumerate}
\item 
I am indebted to Professor X for stimulating discussions.
\item 
However, as we have seen.
\item 
In general.
\item 
It is easily shown.
\item 
To be continued.
\end{enumerate}


\end{comment}



